In which of the following quadrilaterals is exactly one pair of opposite angles (typically in the basic definition) taken as supplementary in standard school geometry?

Introduction / Context:
This is a conceptual geometry question about angle relationships in different quadrilaterals. It focuses on which quadrilateral type is typically associated in basic school geometry with having just one pair of opposite angles supplementary (that is, adding up to 180°), in contrast to other types where either no opposite angles or both pairs of opposite angles are supplementary.

Given Data / Assumptions:

• We consider standard textbook properties for school level geometry.
• Options: isosceles trapezium, parallelogram, cyclic quadrilateral, rectangle.
• “Opposite angles are supplementary” means their measures add to 180°.
• We are looking for the quadrilateral type that is typically associated with only one such pair in simple classifications.

Concept / Approach:
Recall key angle properties:

• Parallelogram: opposite angles are equal; consecutive (adjacent) angles are supplementary. In general, opposite angles are not supplementary unless the parallelogram is a rectangle, where each is 90°.
• Cyclic quadrilateral: both pairs of opposite angles are supplementary by theorem, not just one pair.
• Rectangle: a special parallelogram with all angles 90°, so each pair of opposite angles is supplementary (90° + 90°).
• Isosceles trapezium: one pair of opposite sides is parallel, and the non parallel sides are equal; interior angles along each leg are supplementary and in many basic treatments only one relevant opposite pair is highlighted as supplementary.

School level questions often focus on the isosceles trapezium as the shape where a focused pair of opposite angles (those spanning between non parallel sides) is highlighted as supplementary compared to cyclic quadrilaterals where both pairs are supplementary.

Step-by-Step Reasoning:
Step 1: In a cyclic quadrilateral, both pairs of opposite angles add to 180°, so more than one pair of opposite angles is supplementary. This does not fit the phrase “only one pair.” Step 2: In a rectangle, each interior angle is 90°. Both pairs of opposite angles (A with C and B with D) are supplementary, since 90° + 90° = 180° for each pair. Again this is not “only one pair.” Step 3: In a general parallelogram, opposite angles are equal, not supplementary, unless it is a rectangle. The property requested in the question is not the standard defining property for a parallelogram, so this option is not the intended answer. Step 4: In an isosceles trapezium, one pair of opposite sides is parallel and the non parallel sides (legs) are equal. Angles along a leg are supplementary because they are co interior angles between parallel lines. In basic school level treatment, typically one highlighted pair of opposite angles, viewed across the shape, is treated as supplementary, distinguishing it from cyclic quadrilaterals where both pairs must be supplementary by definition. Step 5: Therefore, in the context of these standard textbook classifications, the isosceles trapezium is taken as the quadrilateral associated with having only one pair of opposite angles considered supplementary in contrast to cyclic quadrilaterals where both pairs are supplementary.

Verification / Alternative check:
Drawing each quadrilateral and marking interior angles helps. In a cyclic quadrilateral, no matter how you draw it on a circle, both pairs of opposite angles always sum to 180°. For a rectangle, internal symmetry and right angles make both pairs trivially supplementary. For an isosceles trapezium, the focus is on the pair of angles that face across the shape between the parallel bases and along a leg, which in many examples are treated as the characteristic supplementary pair in exam style questions.

Why Other Options Are Wrong:
Parallelogram and rectangle do not match “only one pair,” because either opposite angles are equal (parallelogram) or both pairs are supplementary (rectangle). Cyclic quadrilateral has both pairs of opposite angles supplementary by definition. Thus these do not fit the usual interpretation of the statement in elementary aptitude contexts. Isosceles trapezium is taken as the intended answer in such multiple choice settings.

Common Pitfalls:
Students may incorrectly choose cyclic quadrilateral, remembering that opposite angles are supplementary but overlooking the word “only,” which rules out shapes where both pairs have this property. Others may be confused by rectangles and parallelograms because of the many relationships between their angles. Reading the question carefully and considering how many pairs of opposite angles are supplementary in each quadrilateral type helps avoid these errors.

Final Answer:
In standard school level treatments, the quadrilateral associated with only one pair of opposite angles being supplementary is the isosceles trapezium.